dimensionIn in common parlance, the “dimensions” measure of the size of an object, such as a box, area measure of its size, usually given as length, width, and height. In mathematics, the notion of dimension is an extension of the idea that a line is “oneone-dimensional, ” a plane is two-dimensional, and space has is threedimensions-dimensional. In both mathematics and physics one also considers higher-dimensional spaces, such as four-dimensional space-time, where four numbers are needed to characterize a point: three to fix a point in space and one to fix the time. Infinite-dimensionalspacesdimensional spaces, first studied early in the 20th century, have played an increasingly important role , both in mathematics and in parts of physics such as quantum field theory, where they represent the space of possible states of a quantum mechanical system.

In differential geometry one considers curves as one-dimensional, sincea since a single number, or “parameter” parameter, determines a pointon a curve -- for point on a curve—for example, the distance, plus or minus, from a fixed point on the curve. A surface, such as the surface of the Earth, has two - dimensions, since each point can be located by a pair of numbers -- usually numbers—usually latitude and longitude. Higher-dimensional curved spaces were introduced by the German mathematician Bernhard Riemann in 1854 and have become both a major subject of study within mathematicsand mathematics and a basic component of modern physics, from Albert Einstein’s theory of general relativity and the subsequent development of cosmological models of the universe to late-20th-century string superstring theory.

In 1918 , the German mathematician Felix Hausdorff introduced the notion of fractional dimensionwhich . This concept has proved extremely fruitful, especially in the hands of the Polish-French mathematician Benoit Mandelbrot, who coined the word fractaland showed how fractional dimensions could be useful in many parts of applied mathematics.